Quasi-isometry

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The term quasi-isometry is used in mathematics to investigate the “rough” global geometry of metric spaces . It plays an important role in numerous areas of geometry, analysis and geometric group theory , for example in the theory of hyperbolic groups or in proving rigidity theorems.

Definitions

Be and two metric spaces . A (not necessarily continuous) mapping is a quasi-isometric embedding if there are constants and , so that

.

The mapping is called quasi-tight, if a constant exists, so that it for each a are with

A quasi-isometry is a quasi-dense, quasi-isometric embedding.

Two mappings are finite apart if .

The spaces and are called quasi-isometric if there is a quasi-isometric drawing .

Examples

The embedding is a quasi-isometry

Every constrained metric space is quasi-isometric to the point.

The embedding is a quasi-isometry for the Euclidean metric on and . You can set , and in the above definition .

The finite at different generating systems , a group associated Cayley graph are quasi-isometric.

Švarc - Milnor -Lemma: If a finitely generated group acts co- compactly and actually discontinuously through isometrics on a Riemannian manifold , then (the Cayley graph of) is quasi-isometric to . (See also the Švarc-Milnor theorem .)

With one obtains from this in particular: The fundamental group of a compact Riemannian manifold is quasi-isometric to the universal superposition .

properties

  • The identical mapping on a metric space is a quasi-isometry.
  • The concatenation of quasi-isometric embeddings (quasi-isometrics) is again a quasi-isometric embedding (quasi-isometric).
  • An image that has a finite distance from a quasi-isometric embedding (quasi-isometry) is again a quasi-isometric embedding (quasi-isometry).
  • Two metric spaces and are quasi-isometric if and only if there are quasi-isometric embeddings and , so that both and and and are finite apart.

Categories

The metric spaces with the quasi-isometric embeddings form a category according to the above properties . However, this is not of interest for quasi-isometrics, since its isomorphisms must be bijective and therefore many important quasi-isometries are not isomorphisms, such as the quasi-isometry between and mentioned in the examples above .

One therefore moves to a category in which the metric spaces are still the objects, but the morphisms are equivalence classes of quasi-isometric embeddings. Two quasi-isometric embeddings are called equivalent if they have a finite distance, and this obviously defines an equivalence relation. If the equivalence class denotes the quasi-isometric embedding , the definitions result

  • ( Well-defined !)

a category. In this category, the isomorphisms are exactly the equivalence classes of quasi-isometries. The automorphism group of a metric space formed in this category is called its quasi-isometric group.

literature

Individual evidence

  1. ^ Clara Löh: Geometric Group Theory, Springer-Verlag 2017, ISBN 978-3-319-72253-5 , definition 5.1.6
  2. ^ Clara Löh: Geometric Group Theory, Springer-Verlag 2017, ISBN 978-3-319-72253-5 , Proposition 5.1.10
  3. Clara Löh: Geometric Group Theory, Springer-Verlag 2017, ISBN 978-3-319-72253-5 , 5.1.12 and 5.1.13